Filomat

Vol. 31, num. 2, p. 413-423

DOI: 10.2298 / FIL1702413C

Date of publication: 2017-02-27

Abstract:

A subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).]]>

Filomat

Vol. 31, num. 20, p. 6551-6560

DOI: 10.2298/FIL1720551B

Date of publication: 2017

Abstract:

We present a new kind of digraphs, called cyclic Kautz digraphs CK(d, `), which are subdigraphs of the well-known Kautz digraphs K(d, `). The latter have the smallest diameter among all digraphs with their number of vertices and degree. Cyclic Kautz digraphs CK(d, `) have vertices labeled by all possible sequences a1 . . . a` of length `, such that each character ai is chosen from an alphabet containing d + 1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and now also requiring that a1 , a` . Their arcs are between vertices a1a2 . . . a` and a2 . . . aà`+1, with a1 , a` and a2 , a`+1. We give the diameter of CK(d, `) for all the values of d and `, and also its number of vertices and arcs]]>